In this work, we propose a positivity-preserving scheme for solving two-dimensional advection-diffusion equations including mixed derivative terms, in order to improve the accuracy of lower-order methods. The solution to these equations, in the absence of mixed derivatives, has been studied in detail, while positivity-preserving solutions to mixed derivative terms have received much less attent...

We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially nonoscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by [18, 24], a general framework, for arbitrary order of accuracy, ...

We present a positivity preserving numerical scheme for the pathwise solution of nonlinear stochastic differential equations driven by a multi-dimensional Wiener process and governed by non-commutative linear and non-Lipschitz vector fields. This strong order one scheme uses: (i) Strang exponential splitting, an approximation that decomposes the stochastic flow separately into the drift flow, a...

In this paper we consider Runge-Kutta discontinuous Galerkin (RKDG) schemes for Vlasov-Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preservi...

The density and pressure are positive physical quantities in magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP) numerical schemes for ideal compressible MHD is highly desired, but remains a challenge especially in the multi-dimensional cases. In this paper, we develop uniformly high-order discontinuous Galerkin (DG) schemes which provably preserve the positivity of densit...

Ideal MHD equations arise in many applications such as astrophysical plasmas and space physics, and they consist of a system of nonlinear hyperbolic conservation laws. The exact density ρ and pressure p should be non-negative. Numerically, such positivity property is not always satisfied by approximated solutions. One can encounter this when simulating problems with low density, high Mach numbe...

We propose two interpolation-based monotone schemes for the anisotropic diffusion problems on unstructured polygonal meshes through the linearity-preserving approach. The new schemes are characterized by their nonlinear two-point flux approximation, which is different from the existing ones and has no constraint on the associated interpolation algorithm for auxiliary unknowns. Thanks to the new...

A second-order accurate in time, positivity-preserving, and unconditionally energy stable operator splitting scheme is proposed analyzed for reaction-diffusion systems with the detailed balance condition. The designed based on an energetic variational formulation, which reaction part reformulated terms of trajectory, both diffusion parts dissipate same free energy. At stage, trajectory equation...

Concerning positivity, cooperative elliptic and parabolic systems behave like the corresponding equations: a positive source implies that the solution is positive. Systems with a noncooperative coupling do not yield such type of behaviour. For noncooperative elliptic systems there is a restricted, but uniform, positivity result and for the non-cooperative parabolic system there is no positivity...